Convergence of a Robust Deep FBSDE Method for Stochastic Control

Kristoffer Andersson; Adam Andersson; C. W. Oosterlee

doi:10.1137/22M1478057

Convergence of a Robust Deep FBSDE Method for Stochastic Control

Kristoffer Andersson, Adam Andersson, C. W. Oosterlee

Chalmers University of Technology

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

Original language	English
Pages (from-to)	A226-A255
Number of pages	30
Journal	SIAM Journal on Scientific Computing
Volume	45
Issue number	1
DOIs	https://doi.org/10.1137/22M1478057
Publication status	Published - Feb 2023

Bibliographical note

Funding Information:
*Submitted to the journal's Methods and Algorithms for Scientific Computing section February 14, 2022; accepted for publication (in revised form) October 6, 2022; published electronically February 27, 2023. https://doi.org/10.1137/22M1478057 Funding: The work of the first and third authors was supported by the European Union under the H2020-EU.1.3.1. MSCA-ITN-2018 scheme, grant 813261. \dagger Research Group of Scientific Computing, Centrum Wiskunde \& Informatica, Amsterdam, Netherlands ([email protected]). \ddagger Research Group of Computational Mathematics, Chalmers University of Technology and the University of Gothenburg, and Saab AB Radar Solutions, Gothenburg, Sweden ([email protected]). \S Mathematical Institute, Utrecht University, Utrecht, Netherlands ([email protected]).

Publisher Copyright:
© 2023 Society for Industrial and Applied Mathematics.

Funding

*Submitted to the journal's Methods and Algorithms for Scientific Computing section February 14, 2022; accepted for publication (in revised form) October 6, 2022; published electronically February 27, 2023. https://doi.org/10.1137/22M1478057 Funding: The work of the first and third authors was supported by the European Union under the H2020-EU.1.3.1. MSCA-ITN-2018 scheme, grant 813261. \dagger Research Group of Scientific Computing, Centrum Wiskunde \& Informatica, Amsterdam, Netherlands ([email protected]). \ddagger Research Group of Computational Mathematics, Chalmers University of Technology and the University of Gothenburg, and Saab AB Radar Solutions, Gothenburg, Sweden ([email protected]). \S Mathematical Institute, Utrecht University, Utrecht, Netherlands ([email protected]).

Keywords

deep learning
FBSDE
stochastic control

Access to Document

10.1137/22M1478057

andersson-et-al-2023-convergence-of-a-robust-deep-fbsde-method-for-stochastic-controlFinal published version, 961 KBLicence: Taverne

Cite this

@article{2fdf62eed2db4650b9eca4b2f75b5268,

title = "Convergence of a Robust Deep FBSDE Method for Stochastic Control",

abstract = "In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.",

keywords = "deep learning, FBSDE, stochastic control",

author = "Kristoffer Andersson and Adam Andersson and Oosterlee, \{C. W.\}",

note = "Funding Information: *Submitted to the journal's Methods and Algorithms for Scientific Computing section February 14, 2022; accepted for publication (in revised form) October 6, 2022; published electronically February 27, 2023. https://doi.org/10.1137/22M1478057 Funding: The work of the first and third authors was supported by the European Union under the H2020-EU.1.3.1. MSCA-ITN-2018 scheme, grant 813261. \textbackslash{}dagger Research Group of Scientific Computing, Centrum Wiskunde \textbackslash{}\& Informatica, Amsterdam, Netherlands ([email protected]). \textbackslash{}ddagger Research Group of Computational Mathematics, Chalmers University of Technology and the University of Gothenburg, and Saab AB Radar Solutions, Gothenburg, Sweden ([email protected]). \textbackslash{}S Mathematical Institute, Utrecht University, Utrecht, Netherlands ([email protected]). Publisher Copyright: {\textcopyright} 2023 Society for Industrial and Applied Mathematics.",

year = "2023",

month = feb,

doi = "10.1137/22M1478057",

language = "English",

volume = "45",

pages = "A226--A255",

journal = "SIAM Journal on Scientific Computing",

issn = "1064-8275",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "1",

}

TY - JOUR

T1 - Convergence of a Robust Deep FBSDE Method for Stochastic Control

AU - Andersson, Kristoffer

AU - Andersson, Adam

AU - Oosterlee, C. W.

N1 - Funding Information: *Submitted to the journal's Methods and Algorithms for Scientific Computing section February 14, 2022; accepted for publication (in revised form) October 6, 2022; published electronically February 27, 2023. https://doi.org/10.1137/22M1478057 Funding: The work of the first and third authors was supported by the European Union under the H2020-EU.1.3.1. MSCA-ITN-2018 scheme, grant 813261. \dagger Research Group of Scientific Computing, Centrum Wiskunde \& Informatica, Amsterdam, Netherlands ([email protected]). \ddagger Research Group of Computational Mathematics, Chalmers University of Technology and the University of Gothenburg, and Saab AB Radar Solutions, Gothenburg, Sweden ([email protected]). \S Mathematical Institute, Utrecht University, Utrecht, Netherlands ([email protected]). Publisher Copyright: © 2023 Society for Industrial and Applied Mathematics.

PY - 2023/2

Y1 - 2023/2

N2 - In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

AB - In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

KW - deep learning

KW - FBSDE

KW - stochastic control

UR - http://www.scopus.com/inward/record.url?scp=85144952287&partnerID=8YFLogxK

U2 - 10.1137/22M1478057

DO - 10.1137/22M1478057

M3 - Article

AN - SCOPUS:85144952287

SN - 1064-8275

VL - 45

SP - A226-A255

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 1

ER -

Convergence of a Robust Deep FBSDE Method for Stochastic Control

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this